The solution of differential equation
` cos x dy = y (sin x - y ) dx, 0 lt x lt pi //2 ` is

Options

A. ` y = sec x cot x + C cot x`
B. ` cot x = (sec x +C)y`
C. ` y sec x = cot x+C`
D. ` sec x = (tan x + C)y`

Answer

c

Explanation

Given , ` cos x dy = y (sin x - y )dx`
` rArr (dy)/(dx) = (y (sin x - y))/(cos x)`
` rArr 1/(y^(2)) (dy)/(dx) -1/2 tan x = - sec x`
Put ` -1/y = v rArr 1/(y^(2)) (dy)/(dx) = (dv)/(dx)`
` rArr (dv)/(dx) +(tanx) v = - sec x`
` :. IF = e^(int tan x dx) = e^(log sec x) = sec x `
Now, the general soution is given by ,
` v sec x = int - sec^(2) x dx +C_(1)`
` rArr v sec x = - tan x +C_(1)`
` rArr sec x = (C +tan x )y " " [ :' C = -C_(1) and v = (-1)/y]`